Issue 42

O. Krepl et alii, Frattura ed Integrità Strutturale, 42 (2017) 66-73; DOI: 10.3221/IGF-ESIS.42.08

It is evident that the stress description (Fig. 4) by means of the first and the third stress terms ( 1 3 ,    

) is more precise than

the description by the first stress term only ( 1  

 ). It can be seen especially in directions θ = 90° and 270° corresponding to

regions of lower tangential stresses. The stability criterion employing critical value H 1C

following from all three stress terms corresponding to crack initiation into

and the critical values H * 1C

(9), (10) is used. Fig. 5 shows results of the values H 1 matrix, into the inclusion and into the interface. The values H 1

follow from FEM numerical solution of model with unit

are ascertained for unit fracture toughness K IC, m

= 1 MPa·m 1/2 for m = {1, 2,

loading 1 MPa. The critical values H * 1C, m interface}. Particular values H 1C,m

for given fracture toughness of the matrix, the inclusion and the interface are:

* 1C



H

IC, KH m

(13)

m

, m

1C,

Figure 4 : Tangential stress distribution for the case E 1 /E 2 solution on r = 1 mm obtained by one term H 1

= 0.5. Yellow and magenta curve stay for the analytical tangential stress respectively. The cyan curve is the mean tangential stress and H 3

and by two terms H 1

and H 3 . Black dots represent the FEM solution on r = 1 mm. The yellow vertical lines denote the interfaces

obtained by two terms H 1

and black vertical lines the function local and global extreme.

The results show that crack initiation conditions depend on the ratio of materials Young's moduli, and on the fracture toughness of material components (matrix, inclusion) and the interface. In the numerical example E 1 corresponds to Young's modulus of the inclusion. The inclusion here is considered to be a sandstone aggregate with E 1 = 20 GPa. Corresponding fracture toughness of sandstone is between 0.28 and 0.52 MPa·m 1/2 . On the other hand, E 2 corresponds to Young's modulus of matrix. In the numerical example it varies from 26.6 to 80 GPa. The fracture toughness of common hardened cement paste is between 0.1 and 0.8 MPa·m 1/2 , where higher values of E 2 usually match to higher values of K IC,2 . The fracture toughness of interface can widely vary. In case of silicate based composites it is dependent on the manufacturing process and development of interfacial transition zone. The values H 1C,interface are higher than H 1C,1 and H 1C,2 thus it seems that the crack kink to the interface is not probable, but mind that usually K IC,interface is lower than K IC,matrix and K IC,inclusion . When considering particular values of K IC, m for particular ratios E 1 / E 2 , the curves of the critical values H 1C, m for matrix, the inclusion and the interface will change by multiplying as indicated in (13). Thus every particular case will show if crack initiation occurs to matrix, to the inclusion or the interface, i.e. min{ H 1C,1 , H 1C,2 , H 1C,interface }. Let us note that studied material model is universal. Although it supposes sharp concave inclusion embedded in matrix, at the same time it can describe convex corner of stiffer inclusion filled with matrix. In this case E 1 would match to matrix and E 2 to the inclusion. Principally, the approaches will be the same and the results very similar.

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